Looking for a proof in the literature of the following lemma:

Let $K\subset\mathbb{R}^d$ be a bounded domain. Let $P_X$ be a finite dimensional subspace of $\mathcal{H}^k(K)$ then

$$\left\|v\right\|_{\mathcal{H}^k(K)} \leq C diam(K)^{-k} \left\|v\right\|_{L_2(K)}$$

for all $v\in P_X$ and where the constant $C$ does not depend on $diam(K)$.

There is a proof in The mathematical theory of finite element method By Susanne C. Brenner, L. Ridgway Scott p111 but they do not check if the constant $C$ is independent of $diam(K)$.

Thanks in advance.